Building the Calculus Toolkit

Real Analysis · Axiom Academy

LESSON Building the Calculus Toolkit Each differentiation rule — sum, constant multiple, power, product, quotient, chain — derived once from the limit, so you never go back to it. 1. The Foundation: the Limit Definition One definition seeds the entire toolkit. The derivative of a function f at a point x is the limit of the difference quotient — the slope of the secant through (x, f(x)) and as the gap h shrinks to zero. Every rule below is derived the same way: write the combination of functions, feed it into this limit, and simplify using the fact that limits distribute over sums and products . 2. Linearity: Sum, Difference & Constant Multiple Goal: show the derivative respects addition and scaling. Two functions changing together change at the sum of their rates; scaling a function scales its rate. Goal: differentiate x^n in one stroke. Growing x a hair to x + h adds a thin border to the region x^n measures; counting those border pieces gives the rate. For f(x) = x^5 , the rule gives f'(x) = 5x^4 instantly. The animation shows the n = 2 square x^2 : it gains two strips of area as the side grows, and . Goal: differentiate a product fg . Picture the value fg as the area of a rectangle with sides f and g . When x nudges forward, both sides grow — and the new area is the sum of two strips.

This is the written version of the interactive lesson above. See the full Real Analysis course.